On two conjectures of Murthy

TL;DR

This paper advances Murthy's conjectures by improving bounds for complete intersections over algebraic closures of finite fields and establishing new obstructions for splitting projective modules over smooth affine algebras.

Contribution

It improves Mohan Kumar's bound for Murthy's conjecture over _p and introduces an obstruction group for splitting projective modules over certain rings.

Findings

Improved bounds for Murthy's conjecture on complete intersections.

Defined an obstruction group for projective module splitting.

Characterized when a projective module splits based on ideal mappings.

Abstract

This article concerns two conjectures of M. P. Murthy. For Murthy's conjecture on complete intersections, the major breakthrough has still been the result proved by Mohan Kumar in 1978. In this article we improve "Mohan Kumar's bound" when the base field is $\overline{\mathbb F}_p$, and illustrate some applications of our result. Murthy's other conjecture is on a "splitting problem", which is roughly about finding the precise obstruction for a projective $R$-module $P$ of rank $\text{dim}(R)-1$ to split off a free summand of rank one, where $R$ is a smooth affine algebra over an algebraically closed field $k$. Asok-Fasel achieved the initial breakthrough, by settling it for $3$-folds and $4$-folds when $char(k)\neq 2$. For $k=\overline{\mathbb F}_p$ ($p\neq 2$) and $\text{dim}(R)\geq 5$ we define an obstruction group and an obstruction class for $P$ (whose determinant is trivial). As application we obtain: $P$ splits if and only if it maps onto a complete intersection ideal of height $\text{dim}(R)-1$.